The federal government tries to boost small business activities by awarding annu

Question

The federal government tries to boost small business activities by awarding annual grants for projects. All bids are competitive but the chance of receiving a grant is highest if the owner has not received any during the last three years and lowest if awards were given in each of the last three years. Specifically, the probability of getting a grant if none were awarded in the last 3 years is 0.9. It decreases to 0.8 if one grant was awarded, 0.7 if two grants were awarded, and only 0.5 if three were received. (a) Describe this situation as a Markov chain by completing the 1-step transition matrix. Remember to clearly define your states. (b) Determine the expected number of awards per owner per year

Explanation / Answer

If none grant were awarded the probability is 0. 9

If one grant was awarded the probability is 0. 8

If none grant were awarded the probability is 0. 9

If one grant was awarded the probability is 0. 8

If two grants were awarded the probability is 0. 7

If three grants were awarded the probability is 0. 5

One step transition matrix is

0 1 2 3

0 [ 0. 9 0. 8 0. 7 0. 5]

The expected number of awarded per owner per year is E (x)= x.p (x)

x= 0, 1, 2, 3 p (x) = 0. 9, 0. 8, 0. 7, 0.

If none of the grant awarded the probability is 0. 9 = E (x)= x. p (x)= o*0. 9= 0

If one grant was awarded the probability is 0. 8= 1*0. 8= 0. 8

If two grants were awarded the probability 0. 7= 2*0. 7 = 1. 4

If three grants were awarded the probability is 0. 5= 3*0. 5= 1. 5

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